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(/afs/iastate.edu/public/web2c/teTeX/share/texmf/tex/latex/amsfonts/umsb.fd File: umsb.fd 1995/01/05 v2.2e AMS font definitions ) [1] [2] Overfull \hbox (6.78014pt too wide) detected at line 134 [] [] Overfull \hbox (43.4969pt too wide) detected at line 176 [] [] Overfull \hbox (128.26363pt too wide) detected at line 198 [] [] Underfull \vbox (badness 1158) has occurred while \output is active [] [3] Overfull \hbox (7.60947pt too wide) in paragraph at lines 204--215 \OT1/cmr/m/n/12 1.2.\OT1/cmr/bx/n/12 Optimizing a Runge-Kutta for-mula with th e use of Gr^^?obner [] [4] [5] Overfull \hbox (1.64667pt too wide) in paragraph at lines 327--340 []$ [] [6] Overfull \hbox (6.28867pt too wide) in paragraph at lines 379--384 \OT1/cmr/m/n/12 3.2.\OT1/cmr/bx/n/12 Considering Step-Doubling of a (4,4) For- mula as an Eleven- [] Overfull \hbox (1.56458pt too wide) in paragraph at lines 379--384 \OT1/cmr/bx/n/12 Stage Fifth-Order For-mula.[] \OT1/cmr/m/n/12 In an ef-fort to im-prove the (4,4) Runge- [] [7] Overfull \hbox (118.99597pt too wide) detected at line 410 [] [] Underfull \vbox (badness 10000) has occurred while \output is active [] [8] Overfull \hbox (42.25076pt too wide) in alignment at lines 429--441 [][][] [] [] File: ErrVsEff.eps Graphic file (type eps) Overfull \hbox (37.48499pt too wide) in paragraph at lines 449--450 [] [] Underfull \vbox (badness 10000) has occurred while \output is active [] [9] Underfull \hbox (badness 10000) in paragraph at lines 451--452 [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 %*********************************************************** ********[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 % The Newton polynomial interpolant with first derivatives g iven.[] [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 % x is a row n-vector with distinct components, y is a matri x of row[] [] Overfull \hbox (35.2002pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 % n-vectors evaluated at the x components, and dy is a matri x of[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 % first derivative values computed at the corresponding y-va lues.[] [] [10] Overfull \hbox (59.9002pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 % p(x) = c(1) + c(2)(x-x(1)) + c(3)(x-x(1))^2 + c(4)(x- x(1))^2*[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 % *(x-x(2) + ... + c(2*n)(x-x(1))^2*...*(x-x(n-2 ))^2*[] [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 %*********************************************************** ********[] [] Overfull \hbox (47.5502pt too wide) in paragraph at lines 504--504 []\OT1/cmtt/m/n/12 if (m - m2) | (n - n2) | (n - n3) | (one - 1) % Are sizes equal?[] [] Overfull \hbox (35.2002pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 c = NaN; % If not, as sign[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 % NaN to ``c ''[] [] Overfull \hbox (53.7252pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 x = reshape([x; x], 1, 2*n); % Cause x to repeat numbers[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 % Double number of p oints[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 c(:,1) = y(:,1); % Make first column the same as that of y[] [] Overfull \hbox (35.2002pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 % This first FOR loop creates a matrix c with column vec tors[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 % that alternate between y(i)' and y[i,i+1] (c(:,1) = y( :,1))[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 c(:,p+1) = (y(:,p/2+1) - y(:,p/2)) ./ (x(p+1) - x (p));[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 % differences normally (InterpN.m does these when x and y are[] [] Overfull \hbox (10.50018pt too wide) in paragraph at lines 504--504 [] \OT1/cmtt/m/n/12 c(:,p) = (c(:,p) - c(:,p-1)) ./ (x(p) - x(p-k)); [] [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 534--534 []\OT1/cmtt/m/n/12 %*********************************************************** ********[] [] [11] Overfull \hbox (35.2002pt too wide) in paragraph at lines 534--534 []\OT1/cmtt/m/n/12 % This function evaluates a polynomial at c; this polynomial has[] [] Overfull \hbox (35.2002pt too wide) in paragraph at lines 534--534 []\OT1/cmtt/m/n/12 % coefficients from the Newtonian interpolant vector f that were[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 534--534 []\OT1/cmtt/m/n/12 % sampled at value(s) of x. The evaluation is recursive and uses[] [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 534--534 []\OT1/cmtt/m/n/12 %*********************************************************** ********[] [] Overfull \hbox (35.2002pt too wide) in paragraph at lines 534--534 []\OT1/cmtt/m/n/12 elseif (~(m3 - 1) & (n3 > (n2+1))) | (~(n3 - 1) & (m3 > (m2+ 1)))[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 534--534 [] \OT1/cmtt/m/n/12 p(:) = NaN; % If size of x is not at least o ne[] [] Overfull \hbox (4.32518pt too wide) in paragraph at lines 534--534 [] \OT1/cmtt/m/n/12 n = length(f); % Counter spans the length of f[ ] [] Overfull \hbox (35.2002pt too wide) in paragraph at lines 534--534 [] \OT1/cmtt/m/n/12 for k = (n-1):-1:1 % Recursive loop using Horner's Rule[] [] Overfull \hbox (10.50018pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 function [x,y] = RK44Auto(dfun,xspan,y0,c2,c3,AbsTol,RelTol) [] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % dfun string that names a function M-file for derivati ves[] [] Overfull \hbox (16.67519pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % xspan two-component vector -- initial & final values o f[] [] [12] Overfull \hbox (22.85019pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % y solution of DE (Euler approximation) on the mesh x[] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % yex = (25/13)*cos(x) + (5/13)*sin(x) - (25/13)*exp(-5* x);[] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % title('Solution of dy/dx = -5y + 10 cos(x) by Runge-Kutt a')[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % title(sprintf('Maximum error is %8.2e',norm(yex-y,'inf') ))[] [] Overfull \hbox (47.5502pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 if nargin < 7 % Default values for optional para meters[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 if ~((c2-1/2) | (c3-1/2)) % Case for Classic form (c2 = c3 = 1/2)[] [] [13] Overfull \hbox (47.5502pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 b1 = 1/12*(1+6*c2*c3-2*c2-2*c3)/c2/c3; % Formulas for b 's and[] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 b2 = -1/12*(-1+2*c3)/c2/(c2-1)/(c3-c2); % a's determined by[] [] Overfull \hbox (53.7252pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 b3 = 1/12*(2*c2-1)/c3/(c3-1)/(c3-c2); % reduced Grobne r basis[] [] Overfull \hbox (47.5502pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 b4 = 1/12*(6*c2*c3-4*c2+3-4*c3)/(c2-1)/(c3-1); % elements calcu-[] [] Overfull \hbox (10.50018pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 % lated by Maple [] [] Overfull \hbox (53.7252pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 a42 = 1/2*(c2-1)*(-c2-5*c3+4*c3^2+2)/c2/(6*c2*c3-4*c2+3-4* c3)/...[] [] Overfull \hbox (41.3752pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 a43 = -(2*c2-1)*(c2-1)*(c3-1)/c3/(6*c2*c3-4*c2+3-4*c3)/(c3 -c2);[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 [neq one] = size(y0); % Determine # of dependent variabl es[] [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % MAIN LOOP: Take steps one at a time until end of interval reached.[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % First Step: Find first initial stepsize using formula fr om[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % H.A. Watts' paper ``Starting step size for an ODE solver ''[] [] [14] Overfull \hbox (53.7252pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 h = min([h abs(xf-x0)/8]); % h <= 1/8 of integration in terval.[] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % Select step based on error estimate in step just complet ed.[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 % Attempt two RK (4,4) steps and a double step, estimate err or[] [] Overfull \hbox (4.32518pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 k13 = h*feval(dfun,x(n)+c3*h,y(n,:)'+(a31*k11+a32*k12));[ ] [] Overfull \hbox (53.7252pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 k14 = h*feval(dfun,x(n)+c4*h,y(n,:)'+(a41*k11+a42*k12+a43 *k13));[] [] Overfull \hbox (22.85019pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 (b1*k11+b2*k12+b3*k13+b4*k14); % First RK(4,4) st ep[] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 k23 = h*feval(dfun,x(n)+(1+c3)*h,ynp1 +(a31*k21+a32*k22 ));[] [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 k24 = h*feval(dfun,x(n)+(1+c4)*h,ynp1+(a41*k21+a42*k22+a4 3*k23));[] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 (b1*k21+b2*k22+b3*k23+b4*k24); % Second RK(4,4) s tep[] [] [15] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 kd3 = 2*h*feval(dfun,x(n)+2*c3*h,y(n,:)'+(a31*kd1+a32*kd2 ));[] [] Overfull \hbox (29.02519pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 (b1*kd1+b2*kd2+b3*kd3+b4*kd4); % Double RK(4,4) s tep[] [] Overfull \hbox (16.67519pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 Est = norm(ynpd-ynp2,inf) / 30; % 30 = 2^5 - 2[] [] Overfull \hbox (59.9002pt too wide) in paragraph at lines 761--761 [] \OT1/cmtt/m/n/12 if mod(n,32)==0 % Allocate more space for solution[] [] [16] Overfull \hbox (59.9002pt too wide) in paragraph at lines 761--761 []\OT1/cmtt/m/n/12 dy = [feval(dfun,x(n-2),y(n-2,:)'), feval(dfun,x(n-1),y(n-1, :)'),...[] [] Underfull \vbox (badness 10000) has occurred while \output is active [] [17] [18] (outline_6_30.aux) ) Here is how much of TeX's memory you used: 2004 strings out of 10901 22603 string characters out of 71836 79796 words of memory out of 263001 4783 multiletter control sequences out of 10000+0 12063 words of font info for 47 fonts, out of 400000 for 1000 175 hyphenation exceptions out of 1000 34i,8n,32p,295b,393s stack positions out of 300i,100n,500p,50000b,4000s Output written on outline_6_30.dvi (18 pages, 49896 bytes).